{"id":"palomar/PALOMAR-2026-10-16-000009","url":"https://github.com/JD-Jones-ASES/modal-depth/tree/8c5fc2afd424a0fc593c97031ada788befdc5b10","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-16-000009-v1.json","repo":"https://github.com/JD-Jones-ASES/modal-depth","tags":["math.LO","msc:03B45","msc:03F45","modal-logic","interpolation","provability-logic"],"score":0,"title":"Depth-labeled interpolation and substitution in Sacchetti's modal logics","trust":"high","domain":"math.LO","source":"palomar","authors":["JD Jones"],"reviews":0,"subject":"math.LO","summary":"Classifies depth labelings in the modal logics wGL_n for positive n. Every simultaneous formula-valued substitution indexed by atom and depth label preserves theoremhood exactly when the labels satisfy c(k+n)=c(k) at every positive depth k. The interpolation equivalences identify the corresponding labeled property with its ordinary interpolation property and the same periodicity condition; the ordinary interpolation theorems remain explicit inputs. Further results classify period spectra, boxed-atom implications and composition of depth substitutions, and transfer a supplied ordinary…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-16","reactions":{},"formalized":true}